We present a new approach to surface reconstruction in arbitrary dimensions based on the Delaunay complex. Basically, our algorithm picks locally a surface at each vertex. In the case of two dimensions we prove that this method gives indeed a reconstruction scheme. In three dimensions we show that for smooth regions of the surface this method works well and at difficult parts of the surface yields an output well-suited for postprocessing. As a postprocessing step we propose a topological clean up and a new technique based on linear programming in order to establish a topologically correct surface. These techniques should be useful also for many other reconstruction algorithms.
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